簡約 \sqrt[5]{\sqrt[7]{2^{12}}}

解

簡約

5 7 2^{12}

35 2^{12}

十進法表記

1.26826 \dots

解答ステップ

5 7 2^{12}

7 2^{12} = 2 7 2^{5}

= 5 2 7 2^{5}

累乗根の規則を適用する:

n ab = n a n b, a \geq 0, b \geq 0

5 2 7 2^{5} = 52 5 7 2^{5}

= 52 5 7 2^{5}

5 7 2^{5} = 72

= 5272

累乗根の規則を適用する:

n a = a^{\frac{1}{n}}

52 = 2^{\frac{1}{5}}

= 2^{\frac{1}{5}} 72

累乗根の規則を適用する:

n a = a^{\frac{1}{n}}

72 = 2^{\frac{1}{7}}

= 2^{\frac{1}{5}} \cdot 2^{\frac{1}{7}}

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{1}{5}} \cdot 2^{\frac{1}{7}} = 2^{\frac{1}{5} + \frac{1}{7}}

= 2^{\frac{1}{5} + \frac{1}{7}}

\frac{1}{5} + \frac{1}{7} = \frac{12}{35}

= 2^{\frac{12}{35}}

2^{\frac{12}{35}} = 35 2^{12}

= 35 2^{12}